Good question. I usually don't have much leftover, as I scrimp on the extra backing and batting we're 'supposed' to use. Most of the time, it goes in the trash.
However, if you have fabric pieces that are truly wide/long enough to use in a scrap quilt, collect enough to fill a small mesh bag (like for lingerie), put the pieces in the mesh bag and wash the scraps in warm water on a gentle cycle. I've done this and did not have a lot of raveling fabric when I took it out of the lingerie bag. Then separate each piece and put it in the dryer as you normally would.
Otherwise, I don't see why you can't use it as is, except that it might be harder to manage or leave a residue on you needle plate.
I've never washing the with batting scraps, so can't speak to whether that's possible.